Reduction of differential equations

Krystyna Skórnik; Joseph Wloka

ArticleOriginal scientific text

Title

Reduction of differential equations

Authors ¹, ²

Affiliations

Institute of Mathematics, Polish Academy of Sciences, Katowice Branch, Bankowa 14/343, 40-007 Katowice, Poland
Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany

Abstract

Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1)

L y = D^{n} y + a_{n - 1} D^{n - 1} y + ... + a_{0} y = 0

, where

a_{0}, ..., a_{n} \in F

, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system

y_{1}, ..., y_{n}

of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).

Keywords

ENG

differential algebra, linear differential equations, operational calculus

J. E. Humphreys [1] Linear algebraic groups, Springer, Berlin, 1975.
J. E. Humphreys [2] Introduction to Lie algebras and representation theory, Springer, Berlin, 1972.
E. L. Ince [1] Ordinary differential equations, Dover Publ., New York, 1956.
Y. Kaplansky [1] An introduction to differential algebra, Hermann, Paris, 1976.
E. R. Kolchin [1] Algebraic matrix groups and Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1-42.
E. R. Kolchin [2] Differential algebra and algebraic groups, Academic Press, New York, 1973.
S. Lang [1] Algebra, Reading, Addison-Wesley Publ., 1984.
A. R. Magid [1] Lectures on differential Galois theory, American Math. Soc., 1994.
J. Mikusiński [1] Operational calculus, Pergamon Press, New York, 1959.
M. F. Singer [1] Solving homogeneous linear differential equations in terms of second order linear differential equations, Amer. J. Math. 107 (1985), 663-696.
M. F. Singer [2] Algebraic relations among solutions of linear differential equations: Fano's theorem, Amer. J. Math. 110 (1988), 115-144.
M. F. Singer [3] An outline of differential Galois theory, in: Computer algebra and differential equations, E. Tournier (ed.), Academic Press, 1989, 3-57.
K. Skórnik and J. Wloka [1] Factoring and splitting of s-differential equations in the field of Mikusiński, Integral Transforms and Special Functions 4 (1996), 263-274.
K. Skórnik and J. Wloka [2] m-Reduction of ordinary differential equations, Coll. Math. 78 (1998), 195-212.
K. Skórnik and J. Wloka [3] Some remarks concerning the m-reduction of differential equations, Integral Transforms and Special Functions 9 (2000), 75-80.
C. Tretkoff and M. Tretkoff [1] Solution of the inverse problem of differential Galois theory in the classical case, Amer. J. Math. 101 (1979), 1327-1332.
J. T. Wloka [1] Über lineare s-Differentialgleichungen in der Operatorenrechnung, Math. Ann. 166 (1966), 233-256.
A. E. Zalesskij [1] Linear groups, Encycl. of Math. Sciences 37 (Algebra IV), Springer, Berlin, 1993.

Title

Reduction of differential equations

Affiliations

Abstract

Keywords

Bibliography